Video of the talks, lectures and lecture courses

**September 17, 2013 at 17:00, Marble Hall, PDMI, nab. Fontanka 27**

Joint meeting of St. Petersburg Mathematical Society and *Cohomology in Mathematics and Physics* program at Euler Institute

**John W. Morgan**

(Simons Center for Geometry and Physics, Stony Brook University, NY)

**Low dimensional topology: a survey**(video)

**September 23, 2013 at 11:00, Euler Institute, Pesochnaya nab. 10**

Joint meeting of St. Petersburg Mathematical Society and *Cohomology in Mathematics and Physics* program at Euler Institute

**Dennis Sullivan**

(CUNY Graduate Center & Stony Brook University, NY)

**Finite algebraic models of processes in space**(video)

**September 23, 2013, 15:00-15:45, Euler Institute, Pesochnaya nab. 10**

**Nikolai Andreev**

(Steklov Mathematical Institute, Moscow)

**Mathematical études (3D animated films)- have fun!**

**September 23, 2013 at 17:00, Marble Hall, PDMI, nab. Fontanka 27**

Joint meeting of St. Petersburg Mathematical Society and *Cohomology in Mathematics and Physics* program at Euler Institute

**James H. Simons**

(Stony Brook University, NY)

**Roots of differential cohomology** (video)

**September 24, 2013 at 11:00, Euler Institute, Pesochnaya nab. 10**

**Ludwig D. Faddeev**

**Anomalies and cohomologies** (video)

Anomaly is an object in the quantum theory of fields, physicists like fancy words. Field Theory is a system with infinite number of degrees of freedom and upon quantisation some symmetries break down. This is called anomaly.

I will argue that in the case of the Yang-Mills theory the symmetry is not broken but is rather modified. Group action is supplied with a factor and/or is substituted by a projective presentation. Corresponding one and two cocycles are produced by the Chern-Simons descent procedure starting from the third Chern class.

**September 24, 2013 at 12:00, Euler Institute, Pesochnaya nab. 10**

**James Simons & Dennis Sullivan**

**Differential K-theory and its Characters** (video)

One begins with how complex bundles with connections define, via the Chern-Simons equivalence relation on connections, the extension (called differential K-theory) of usual K-theory by all total odd forms modulo pull backs from the unitary group of the canonical primitive closed forms...

One also defines the analog of differential characters for these objects. A K-character assigns to each odd dimensional closed manifold with additional geometrical structure mapping into the base a number in R/Z. These numbers satisfy a deformation property and a product property.

The first theorem gives a bijection between K-characters and the elements of differential K-theory. One corollary is a differential geometric construction of the known (but unpublished) set of complete numerical invariants for elements in usual complex K-theory.

The first theorem also yields a natural push forward in differential K-theory for families of almost complex manifolds geometrized as a riemannian submersion. A non trivial computation shows that the natural product connection on the total space used to define the push forward is chern-simons equivalent to the limit of the levi civita connections on the total space as the fibre are scaled down to points.

Finally the stage is set to apply the Atiyah-Patodi-Singer theorem to obtain an analytic computation of the differential K-theory push forward at each value of the K-character as a limit of eta-invariants.

**September 30, 2013, at Polit.ru**

**Leon Takhtajan**

Профессор Леон Тахтаджян расскажет о математике как форме существования

**Polit.ru public lecture, October 3, 2013 at 19:00, cafe ZaVtra, Moscow**

**Leon Takhtajan**

**"Математика как форма существования мира идей в нашем сознании"** (video), (video and full transcript)

(four-six lectures, 90 min = 2 x 45 min each)

- Course 1. “Cohomology: differential, etale and motivic cohomology theories” – I. Panin (video)
- Course 2 . “Chern-Simons theory, knot homology and quantum curves” – S. Gukov (video)
- Course 3. “Introduction to topological quantum field theories: Seiberg-Witten and Donaldson-Witten theories” – M. Marino & A. Losev (video)
- Course 4 “Mirror symmetry: quantum cohomology, Landau-Ginzburg models & D-branes” – D. Orlov (video)

(two-four lectures, 90 min = 2 x 45 min each)

- Topics 1 “Quantum invariants” – R. Kashaev (video)
- Topics 2 “Introduction to hyperkahler manifolds” – M. Verbitsky (video)
- Topics 3 “Geometric representation theory” – H. Nakajima (video)
- Topics 4 “Associators, Grothendieck-Teichmüller group and flat connections” – A. Alekseev (video)
- Topics 5 “Algebraic integrable systems and quantum Riemann surfaces” – I. Krichever & F. Smirnov (video)
- Topics 6 “Supersymmetric gauge theories and quantum integrability” – V. Pestun (video)
- Topics 7 “Quasi-maps, Uhlenbeck spaces and quantum K-theory” – A. Braverman (video)

- "Introduction to Riemannian geometry, curvature and Ricci flow, with applications to the topology of 3-dimensional manifolds" – J. Morgan (video)
- "Kahler-Einstein metrics and Ricci flow" – G. Tian (video)

S. Gukov, R. Kashaev, I. Krichever, A. Losev, M. Marino, J. Morgan, H. Nakajima, D. Orlov, I. Panin, V. Pestun, J. Simons, F. Smirnov, D. Sullivan, G. Tian, M. Verbitsky.

**For lecture courses**

**Marcos Marino, An introduction to Donaldson-Witten theory**

**Marcos Marino, Chern-Simons Theory, Matrix Models, and Topological Strings**

**Sergei Gukov, Quantization via mirror symmetry**

**Sergei Gukov and Ingmar Saberi, Lectures on knot homology and quantum curves**

**Hiraku Nakajima, Geometric representation theory I**

**Hiraku Nakajima, Geometric representation theory II**

**Hiraku Nakajima, Geometric representation theory III**

**John Morgan Topology of manifolds**

**John Morgan and Gang Tian, Ricci flow and Poincare conjecture**

**Misha Verbitsky, Hyperkahler manifolds. Lecture 1**

**Misha Verbitsky, Hyperkahler manifolds. Lecture 2**

**Misha Verbitsky, Hyperkahler manifolds. Lecture 3**

**Books on quantum mechanics and quantum field theory**

**L.D. Faddeev and O.A. Yakuboskii, Lectures on Quantum Mechanics for Mathematics Students**

(Л. Д.Фаддеев, О.А.Якубовский, *Лекции по квантовой механике для студентов-математиков*, НИЦ Регулярная и хаотическая динамика, Ижевск, 2001)

**L.A. Takhtajan, Quantum Mechanics for Mathematicians**

(Л.А. Тахтаджян, *Квантовая механика для математиков*, НИЦ Регулярная и хаотическая динамика, Ижевск, 2011)

**L.D. Faddeev and A.A. Slavnov, Gauge Fields: An Introduction To Quantum Theory**

(A.A.Славнов, Л. Д.Фаддеев, *Введение в квантовую теорию калибровочных полей*, Наука, М., 1988)

**P. Deligne and others, Eds, Quantum Fields and Strings: A Course for Mathematicians**

26 - 28 September 2013

3 - 5 October 2013

14 - 18 October 2013